We know that if $M$ is a $d-$dimensional manifold, then $H_n(M, M\setminus \{x\})=\mathbb{Z}$ for $n=d$ and equals to $0$ else.
I can understand formally the result and its proof, but I can't understand what it is actually saying. What is the intuitive meaning of this result?
Thank you in advance
If you understand the proof you probably understand the intuition. Only that part of $M$ in a neighbourhood of $x$ is important, since we have relative to $M-\{x\}$. And since $M$ is a manifold it looks locally like $\mathbb{R}^d$ so it says that regardless of how $M$ look globally, $$H_n(M,M-\{x\}) =H_n(D,D-\{x\})$$ where $D$ is a $d$-disk. This is just a paraphrase of the proof.
